Problems: Simply Connected Regions 1. Which of the regions shown below are simply connected? (a) (b) (c) Answer: Region (a) is not simply connected – the “puncture” at the center of the disk would prevent any simple closed curve around it from contracting to a point while remaining within the region. Regions (b) and (c) are simply connected. We frequently “cut” a region like (a) to create a simply connected region similar to (b). Region (c) illustrates the fact that simply connected regions aren’t always simple! For each of the vector ﬁelds described below, ﬁnd the domain on which it is deﬁned and continuously diﬀerentiable. Is that domain simply connected? 2. sin(x2 + y 2 )i + cos(x2 + y 2 )j Answer: The vector ﬁeld is deﬁned and diﬀerentiable at all points (x, y). This region is simply connected. 3. |x|i + 0j Answer: This vector ﬁeld is not diﬀerentiable when x = 0. The region is the union of the open left and right half-planes and is not simply connected. xi + yj x2 + y 2 Answer: The region is the punctured plane (x, y) = (0, 0). It is not simply connected. 4. yi − xj y2 Answer: The region is the union of the upper and lower half planes; y = 0. It is not simply connected. 5. 6. x2 − 1 i + y 2 − 1j Answer: This vector ﬁeld is deﬁned and continuously diﬀerentiable when x2 and y 2 are greater than 1. This region is the plane minus a square of side length two and is not simply connected.